Let \(InfPath^{\mathcal {SM}}_{\pi }\)  (resp., \(FinPath^{\mathcal {SM}}_{\pi }\) ) be the set of infinite (resp., finite) paths starting from \(s_0=s^I\)  in the SMDP \(\mathcal {SM}\)  under a policy \(\pi \) . The time-abstract behavior of the SMDP \(\mathcal {M}\)  under a policy \(\pi \)  is defined on a probability space \((InfPath^{\mathcal {SM}}_{\pi }, \mathcal {F}_{InfPath^{\mathcal {SM}}_{\pi }}, Pr^{\mathcal {SM}}_{\pi })\)  [1]}.
